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Ellipsoidal Harmonics by George Dassios

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8Connection between sphero-conal and ellipsoidal harmonics

8.1 Geometrical reduction

It is of interest to realize that, at the geometrical level, an ellipsoid deforms continuously to a sphere, but at the analytical level, the ellipsoidal harmonics are not reducible in a straightforward and unique way to the corresponding spherical harmonics. In other words, the kernel space of the ellipsoidal Laplacian does not degenerate in a unique way to the kernel space of the spherical Laplacian without disturbing the orientation of the spherical reference system. That is, if we choose to reduce the ellipsoid first to the prolate spheroid and then to the sphere, we end up with a polar axis that is different to the case where we first reduce the ellipsoid ...

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